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Properties

Label 2-1120-56.19-c1-0-18
Degree $2$
Conductor $1120$
Sign $0.415 + 0.909i$
Analytic cond. $8.94324$
Root an. cond. $2.99052$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.90 + 1.10i)3-s + (0.5 − 0.866i)5-s + (−0.584 − 2.58i)7-s + (0.922 − 1.59i)9-s + (2.90 + 5.03i)11-s − 4.83·13-s + 2.20i·15-s + (3.78 − 2.18i)17-s + (−1.63 − 0.945i)19-s + (3.95 + 4.27i)21-s + (0.157 + 0.0911i)23-s + (−0.499 − 0.866i)25-s − 2.54i·27-s − 4.38i·29-s + (−2.03 − 3.53i)31-s + ⋯
L(s)  = 1  + (−1.10 + 0.635i)3-s + (0.223 − 0.387i)5-s + (−0.220 − 0.975i)7-s + (0.307 − 0.532i)9-s + (0.875 + 1.51i)11-s − 1.34·13-s + 0.568i·15-s + (0.917 − 0.529i)17-s + (−0.375 − 0.216i)19-s + (0.862 + 0.933i)21-s + (0.0329 + 0.0189i)23-s + (−0.0999 − 0.173i)25-s − 0.489i·27-s − 0.815i·29-s + (−0.366 − 0.634i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.415 + 0.909i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7865937453\)
\(L(\frac12)\) \(\approx\) \(0.7865937453\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.584 + 2.58i)T \)
good3 \( 1 + (1.90 - 1.10i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.90 - 5.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.83T + 13T^{2} \)
17 \( 1 + (-3.78 + 2.18i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.63 + 0.945i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.157 - 0.0911i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.38iT - 29T^{2} \)
31 \( 1 + (2.03 + 3.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.69 + 2.13i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.44iT - 41T^{2} \)
43 \( 1 - 2.10T + 43T^{2} \)
47 \( 1 + (-0.946 + 1.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.54 + 4.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.35 + 3.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.89 + 8.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.04 + 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.80iT - 71T^{2} \)
73 \( 1 + (-7.34 + 4.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.5 - 6.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.9iT - 83T^{2} \)
89 \( 1 + (5.11 + 2.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.846178878843460759449179280914, −9.329031257815524822553739173022, −7.79481902513001779603272596880, −7.11511189612369618469265697095, −6.29341107521925757238625508528, −5.13713643975708800866457146948, −4.65728473822040001385394339505, −3.82196588060896534075956618755, −2.08393626518413426448319023063, −0.45002829086945837346214466292, 1.17935186170772674946317938014, 2.61227913047776763424040116181, 3.70581564017759398918346937772, 5.38889851345338528882939978011, 5.66961483642704083653789090142, 6.55590535604139184251383279403, 7.18128616699109086420183326482, 8.414533057584798398481820732715, 9.099140173979723973160453044628, 10.12782104279799635276315792242

Graph of the $Z$-function along the critical line